The formation and propagation of cracks in industrial structures represent a major safety issue. In addition to the visual and aesthetic aspects, the initiation and propagation of cracks can lead to undesirable leaks in sensitive structures such as, for example, the containments of a nuclear reactor, a hydroelectric dam, a cryogenic tank, or piping.
An understanding of the fracture mechanics of materials used in such structures and a quantification of the cracking phenomenon are therefore of considerable interest.
The cracking behavior of a structure composed of a material is generally related to its state of strain and to the mechanical stresses near the crack tip. This local state of stresses and strains can be modified by the application of displacements at the edges of the structure and of mechanical forces such as external pressure, the weight of the structure, or a centrifugal force to which the structure is subjected. To generalize these concepts in the technical terminology commonly used in the literature, these are called mechanical “loads” sustained by the structure. The term “load” as used below covers all these concepts.
Typically, the cracking of a material is determined based on experiments intended to apply to a sample of a volume of material called a specimen, in the laboratory, the same mechanical loads as those sustained by the volume of material during use. It is thus common to apply pre-established mechanical loads and to observe the evolution of a crack propagating in the volume of the material. A numerical model based on a finite element calculation can be used to simulate the expected crack in the volume after applying previously established mechanical loads to the material. A comparison of experiments conducted on specimens in simulations refines the numerical model and provides qualitative and/or quantitative information on the nature of the cracking behavior of the material.
Models of crack propagation in a material are generally based on two different approaches to modeling the response of a volume of material to a mechanical load.
One approach, called “linear elastic fracture mechanics” (LEFM), provides a theoretical framework suitable for materials exhibiting an underlying linear behavior. The crack presents as a geometrical singularity of well-established dimension. LEFM is based on a general calculation of physical parameters of the volume of material that can be used to characterize the stress and strain field around the crack tip. These parameters are generally referred to as “stress intensity factors” (SIF), and are known in the field of Engineering specializing in industrial calculations. These parameters, generally denoted KI, KII, KIII according to the nature of the associated stress, can be related to the terms σij of the stress field by the following relation:
      s    ij    =                    K        m                              2          ⁢          pr                      ⁢                  f        ij            ⁡              (        q        )            where r and θ are polar coordinates near the crack tip and f denotes a function of the angle θ. The indices I, II and III accompanying the SIF (Km) respectively denote three cracking modes observed in the materials. Mode I corresponds to a tensile load perpendicular to a plane containing the crack, mode II corresponds to a shear load acting in parallel to a plane containing the crack and perpendicular to the crack front, and mode III corresponds to a shear load acting in parallel both to a plane containing the crack and to the crack front.
Typically, the SIF values evolve linearly as a function of the mechanical load applied to the material. These values can be determined by analyzing the stress state locally near the crack tip. However, this linear behavior is only observed for loads below a critical fracture load, associated with threshold SIF values called the toughness.
“Near the tip” as discussed above generally refers to a zone around the crack tip, typically a toric or spherical area, in which finite element calculations are performed in order to determine the loading response of the material. By analogy, an empirical representation of the cracking behavior of a material generally involves introducing a zone around the tip of the crack called the “process zone”. This zone includes the points of the material which are most susceptible to cracking under the applied mechanical load.
In LEFM, a crack is initiated when a threshold value for the SIFs is reached. This threshold value is an intrinsic property of the material. Application of LEFM generally assumes a defined gridding for the volume of material in order to establish a linear algebraic equation for solving the system linking the load terms to the SIFs. One shortcoming of this approach, which does not take into account the evolution of the process zone, is that it is essentially adapted for modeling crack propagation in brittle materials such as glass, ceramics and refractory materials, or plexiglas, but fails to correctly model crack propagation in materials of greater non-linearity (plasticity, micro-cracking) such as concrete, graphite, or metal materials at low temperatures.
Another class of approaches, which fills this gap, is based on modeling the average field of damage of the volume of material, and is called the “diffuse damage model.” This approach uses a thermodynamic description of the volume of material and of the appearance of defects in this volume, taken into account as an average value. Application of a load to the volume is expressed in this model by a certain probability of transitioning from a healthy elastic behavior to a softening behavior, characterized by an irreversible drop in mechanical stiffness. The birth and growth of microscopic defects are revealed by the evolution of the field of damage. The approach of the diffuse damage model thus enables describing cracks as contour lines in the field of damage, but scalar. No regridding of the volume is required and the damage at a point of the volume is determined in relation to the mechanical load state near that point.
However, these two approaches both have limitations for properly modeling the cracking behavior of a specimen. In particular, both approaches have difficulty modeling crack propagation in a specimen subjected to mechanical loads defined beforehand as described above. On the one hand, there is a problem in identifying the model parameters. Furthermore, the cracking models are based on certain approximations and do not reproduce the actual cracking behavior of a material with sufficient accuracy. In particular, the propagation of a crack in the volume of a material subjected to a predefined mechanical load typically does not follow a linear behavior. Non-linearities are not compatible with LEFM, and also return unsatisfactory results with the diffuse damage approach. Lastly, the appearance of nonlinear cracking behavior does not meet industrial specifications in terms of controlling cracking phenomena.
A method for obtaining more refined and accurate quantitative information on the cracking behavior of a material is therefore desired.
An original approach which increases the reliability of the two approaches outlined above is proposed in the document “Drawing with a crack in concrete: a hybrid test to control mixed-mode crack propagation” by Carpiuc et al, 21st Congrès Français de Méchanique (French Mechanics Conference), 2013. According to the original approach proposed in that document, it is not a question of applying a fixed mechanical load to a volume of material and observing the crack which then propagates in the material. The method proposed in that document consists of establishing a target crack path in the volume of material and successively applying different mechanical loads in order to propagate a crack in a specimen along the target crack path. The mechanical loads applied to the specimen are reevaluated throughout the cracking process, after each increment propagating the crack in the volume, to ensure that the crack created in the specimen does indeed follow the target crack path in a controlled manner. This original approach has the advantage of dividing the crack development into several steps, which reduces the nonlinearities of each step in the propagation of a crack in a specimen and increases the reliability of predictive models used to simulate the propagation of the crack. In addition, this approach allows using a propagation model based on LEFM to successively simulate the propagation of a crack which is redirected within the volume of material so that it follows the target crack path.
However, despite the improvements provided by this approach to characterizing the cracking behavior of a material, it still does not always permit controlling the speed of crack propagation and thus avoiding the appearance of instabilities during crack propagation.
A method is therefore sought for controlling the cracking of a material by generating a crack along a target path, in a stable and controlled manner.